Aug 25 Continuity

Simple Definition

This is a non-mathematical, non-academic definition of continuous: you can draw your function without lifting your pencil.

Discontinuous

Continuous

Actual Mathematical Definition

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Recall that this is the definition of a limit:
$$\forall \epsilon > 0, \exists \delta > 0 \text{ s.t. if } 0 < ||x-c|| < \delta \to ||f(x) - f(c)|| < \epsilon$$
You might see it written out differently, but the definition of continuous is very similar
$$\forall \epsilon > 0, \exists \delta > 0 \text{ s.t. if } ||x-c|| < \delta \to ||f(x) - f(c)|| < \epsilon$$
The only difference is the $0
Now, think about the continuous definition.
If we consider a super small $\epsilon$, then if $x = x_0$, which less than any $\delta > 0$, the expected f(x) values and the jump point will be greater than epsilon.

Examples

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Prove the heaviside function is not continuous at x = 0.
$$f(x) = \begin{cases}1 & x \ge 0 \\0 & x \lt 0\end{cases}$$
Now, we pick an $\epsilon$. We can show this doesn't work for any $\epsilon < 1$. Let's say $\frac{1}{2}$.
$$\text{Let } \epsilon = \frac{1}{2}$$
Now, we have to prove it for all $\delta > 0$, so we say
$$\text{Let } \delta > 0 $$
We now have a ball around 0. We want to show that a point x is within that ball, but the difference between that and $f(0) > \frac{1}{2}$. So, we have to pick a negative x value.
$$\text{Let } x = -\frac{\delta}{2}$$
$$||x - 0|| = \frac{\delta}{2} < \delta$$
$$||f(x) - f(0)|| = ||0 - 1|| = 1 > \dfrac{\epsilon}{2}$$
Now, we know there exists an $\epsilon > 0$ ($\frac{1}{2}$) such that for every $\delta > 0$, there exists an x that is within $\delta$ of 0 and f(x) is not within $\epsilon$ of $f(0)$.

A Function is Continuous

That means that the function is continuous at every point in its domain.

Differentiability Implies Continuity

We will prove this statement in two steps. First, we show the Lipschitz Condition is true. Now, once we show that, we can show that it’s continuous.

Lipschitz Continuity

In fact, if a function satisfies the Lipschitz condition, it is considered Lipschitz continuous. An easy way to see if something is Lipschitz continuous is if the derivative is bounded. Why is that? Well the Lipschitz condition states that you can create a line that is greater than or equal to the function. If the derivative is unbounded, then the function will exceed that line.